WBK方程、AKNS方程及分数阶对流扩散方程的新解

发布时间：2018-03-03 18:11

本文选题：Hirota双线性方法　切入点：广义WBK方程组　出处：《渤海大学》2017年硕士论文　论文类型：学位论文

【摘要】：随着孤子理论和科学技术的不断发展,近年来求解孤子方程的精确解已成为孤子理论研究中的重中之重.在孤子方程的诸多求解方法中,Hirota双线性方法起着举足轻重的作用,是众多学者的研究焦点,Hirota双线性方法属于构造性求解,这种构造性求解法较于其他求解法的优势在于其不依赖于方程的Lax对或者谱问题,正是由于这种构造性求解法简捷、直观的特点,激发了学者们的研究热潮.近几年来分数阶问题也引起了人们的广泛关注,分数阶非线性偏微分方程成为研究热题.本文一方面利用Hirota双线性方法来分别构造广义WBK方程组和广义AKNS方程族的单孤子解、双孤子解以及N-孤子解的表达式.另一方面紧紧围绕分数阶微积分、分数阶导数的相关知识,构造带有初边值条件的变系数时间分数阶对流扩散方程的新的精确解.本文的主要工作概括如下:首先,在第三章和第四章中推广并应用Hirota双线性方法构造广义WBK方程组和广义AKNS方程族的单孤子解、双孤子解、三孤子解,并归纳出N-孤子解的表达式,这得益于成功将广义WBK方程组和广义AKNS方程族分别进行转化,在广义WBK方程组和广义AKNS方程族的求解中,关键是通过一系列有效变换找到它们的双线性形式,从而获得新的孤子解.其次,在第五章中运用分离变量法和Mittag-Leffler函数的性质获得一类变系数时间分数阶对流扩散方程的满足一定初边值条件的精确解的统一表达式,进而通过考虑这类对流扩散方程的具体实例和具体初边值条件得到新的分离变量解,这为求解分数阶的非线性偏微分方程提供重要参考价值.
[Abstract]:With the development of soliton theory and science and technology, the exact solution of soliton equation has become the most important part in the study of soliton theory in recent years, and Hirota bilinear method plays an important role in solving soliton equation. Hirota bilinear method is the focus of many scholars. The advantage of this method over other methods is that it does not depend on the Lax pair of equations or spectral problems. In recent years, the problem of fractional order has also aroused widespread concern. The fractional order nonlinear partial differential equation has become a hot topic. In this paper, on the one hand, we use Hirota bilinear method to construct the single soliton solutions of generalized WBK equations and generalized AKNS equations, respectively. Expressions of double soliton solutions and N-soliton solutions. On the other hand, the knowledge of fractional calculus, fractional derivative, A new exact solution of time fractional convection-diffusion equation with variable coefficients with initial boundary value condition is constructed. The main work of this paper is summarized as follows: first of all, In chapter 3 and chapter 4th, we generalize and apply Hirota bilinear method to construct the single soliton solution, double soliton solution, three-soliton solution of generalized WBK equations and generalized AKNS equation family, and generalize the expression of N-soliton solution. This is due to the successful transformation of the family of generalized WBK equations and generalized AKNS equations. In the solution of generalized WBK equations and generalized AKNS equations, the key is to find their bilinear forms through a series of effective transformations. A new soliton solution is obtained. Secondly, in Chapter 5th, by using the method of separating variables and the properties of Mittag-Leffler function, a unified expression of exact solutions of a class of fractional convection-diffusion equations with variable coefficients is obtained, which satisfies certain initial boundary value conditions. By considering the concrete examples and the concrete initial boundary value conditions of this kind of convection-diffusion equation, a new solution of separated variables is obtained, which provides an important reference value for solving fractional nonlinear partial differential equations.
【学位授予单位】：渤海大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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